Dynamic Model
Non-linear differential equations are used to model the behavior of mechanical and electrical systems with non-linear stiffness characteristics. The Duffing oscillator describes a system featuring a cubic stiffness term that represents hardening or softening structural springs. This mathematical model helps design engineers predict jump phenomena and chaotic movements in MEMS resonators and energy harvesters.
It acts as a fundamental framework for studying complex dynamics in micro-scale devices.
Resonant Behavior
Stiffness changes as the displacement of the resonator increases under dynamic loads. When the excitation frequency is swept across the resonance region, the output amplitude exhibits a sudden jump at a specific point. This jump creates a region of bistability where multiple states exist for a single frequency.
Managing this transition is essential for maintaining sensor sensitivity in non-linear regimes.
Operational Limit
Phase space trajectories become complex when the system is driven into chaotic regimes. These chaotic oscillations lead to unpredictable frequency responses and signal degradation.
Experimental Identification
Frequency response testing identifies the cubic stiffness parameter by tracking the hysteretic jumps during upward and downward frequency sweeps. High-resolution laser vibrometers measure the physical velocity of the resonator while a signal analyzer drives the electrostatic actuator. The obtained response curves are fitted to numerical solutions of the non-linear equation to extract the exact coefficients.
This characterization helps production teams adjust the calibration curves of individual sensor shipments.
